between homotopy classes of continuous maps X→𝒦(V)X \to \mathcal{K}(V) and equivalence classes of the group completed stackification of 2-vector bundles,

where the colimit is over acyclic Serre fibrations (Note: these are not acyclic fibrations in the usual sense, rather their fibres have trivial integral homology) and Gr(−)Gr(-) indicates the Grothendieck group completion using the monoid structure arising from the direct sum of 2-vector bundles.

(R,⊗,1)(R , \otimes, 1) is a monoidal category, assumed to be strict monidal in the following;

a distributivity law.

Examples

E=CoreE = CoreFinSet, the core of the category of finite sets and morphisms only between sets of the same cardinality.

In the skeleton, objects are natural numbers n∈mathbNn \in \mathb{N}, ⊕\oplus and ⊗\otimes is addition and multiplication on ℕ\mathbb{N}, respectively. Here c⊕c_{\oplus} is the evident natural isomorphism between direct sums of finite sets.

V=CoreV = CoreVect the core of the category of finite dim vector spaces, with morphisms only between those of the same dimension.

…

Definition For RR a bimonoidal category, write Matn(R)Mat_n(R) for the n×nn \times n matrices with entries morphisms in RR. Then matrix multiplication is defined using the bimonoidal structure on RR. This gives a weak monoid structure.

Let Gln(R)Gl_n(R) be the category of weakly invertible such matrices. This is the full subcategory of Matn(R)Mat_n(R). We get a diagram of pullback squares

Notice that for HRH R to be a spectrum we only need the additive structure (R,⊕,0,c⊕)(R, \oplus, 0, c_{\oplus}). The point is that the other monoidal structure ⊗\otimes indeed makes this a ring spectrum. This is a not completely trivial statement due to a bunch of people, involving Peter May and Elmendorf-Mandell (2006).

Examples

For the category R:=E=Core(FinSet)R := E = Core(FinSet) of finite sets as above we have that HEH E is the sphere spectrum.

For R:=V=Core(FinVect)R := V = Core(FinVect) the core of complex finite dimensional vector spaces we have HVH V is the complex K-theory spectrum.

For VℝV_{\mathbb{R}} analogously we get the real K-theory spectrum.

So by the above theorem

𝒦(E)≃K(S)≃A(*)\mathcal{K}(E) \simeq K(S) \simeq A(*)

𝒦(V)≃K(ku)\mathcal{K}(V) \simeq K(ku);

etc.

Remarks

A. Osono: The equivalence 𝒦(R)≃K(HR)\mathcal{K}(R) \simeq K(H R) of topological spaces is even an equivalence of infinity loop space?s;